ÉCOLE DE PHYSIQUE DES HOUCHES STABILITY OF FLUID FLOWS Carlo COSSU Institut de Mécanique des Fluides Toulouse CNRS Université de Toulouse, also at Département de Mécanique, École Polytechinique http://www.imft.fr/carlo-cossu/ http://www.enseignement.polytechnique.fr/profs/mecanique/carlo.cossu/ Le futur de l'astrophysique des plasmas 25/2-8/3/2013
WELCOME! LECTURE CONTENTS: * Introduction to hydrodynamic stability * Linear stability: method of normal modes * Transient energy growth in stable systems Some examples/results given (only) for hydrodynamic instabilities in incompressible shear flows
INTRODUCTION
An example of transition to turbulence Transition to turbulence turbulent flow: multi-scale, non-periodic, unpredictable instabilities develop: the flow loses symmetry & is unsteady laminar flow: maximum symmetry predictable, usually steady
Some questions Laminar flow solution: same symmetries of problem Why not often observed? Why observed solutions are less symmetric than the problem data? (e.g. unsteady if the problem is steady or non axi-sym etc.) Can we predict when steadiness and symmetry are lost and why?
The hydrodynamic stability main idea Solutions can be observed only if they are stable!
AN EXAMPLE: KELVIN-HELMHOLTZ INSTABILITY
Mechanism: induced forces amplify perturbations basic flow: counter streaming flows with opposite velocities U here the interface has been perturbed sinusoidally H In first approximation: h u = const. (mass conservation) p+ρu2/2=const (Bernoulli) h<h u > U p<p h>h u < U p>p
increasing time The tilting tank experiment two immiscible fluids in a tilting tank. lighter: transparent, heavier: coloured tilted tank: lighter fluid pushed up, heavier down counter-streaming flows for finite time
Billow clouds near Denver, Colorado, (picture by Paul E. Branstine. meteorological details, in Colson 1954) source: Drazin 2001
Instability active also in turbulent flows Increasing Reynolds number Instability of laminar flow Transitional flow Brown & Roshko, J. Fluid Mech. 1974 Large scale coherent structures in turbulent flow
DEFINITIONS OF STABILITY
Evolution equations, basic flow, perturbations evolution equation state vector basic state / perturbation decomposition parameter(s) perturbation basic state perturbations evolution equation obtained replacing the decomposition in the evolution eqn. & then removing the basic state evolution eqn.
Norm of the perturbations need to define a scalar perturbation amplitude e.g. energy max absolute value other definitions OK as long as they are norms:
Stability definitions Lyapunov Asymptotic (standard definition used in the following) Unconditional Remarks: * single initial condition is sufficient to prove instability of the base flow * to prove stability one has to prove it for ALL possible initial conditions
Dependence on the control parameter stability properties depends on the control parameter r norm of initial perturbation δ no relaxation to basic flow for at least one initial condition relaxation to basic flow for all ICs rg unconditional stability conditional (in)stability critical r r rc unconditional instability shown typical case with rg < rc but can e.g. happen that rc is infinite or that rg = rc
Linearized equations start from perturbation eqns Taylor expansion of r.h.s. near basic flow replace & neglect higher order terms linearized evolution equations linearized (tangent) operator (Jacobian)
Linear stability analysis linear problem decomposition of generic solution on basis of `fundamental' solutions constants depend on the initial condition complete set of linearly independent solutions linear instability if at least one fundamental solution is unstable linear stability if ALL fundamental solutions are stable
new problem: compute the basis of linearly independent solutions and analyze their stability general method available for steady basic flows (steady L): method of normal modes
THE METHOD OF NORMAL MODES: LINEAR SYSTEMS OF ODEs
System of autonomous ODEs Consider a system of N 1st-order ordinary differential equations with constant coefficients: state: N-dimensional vector Linear operator: NxN matrix L does not depend on t!
Eigenvalues & eigenvectors eigenvector (direction left unchanged by L) eigenvalue (scalar) Homogeneous system: non-trivial solutions if characteristic eqn algebraic N-th order N roots for s N eigenvalue-eigenvector pairs
Modal decomposition for ODEs assume N distinct eigenvalues N linearly independent eigenvectors = eigenvector basis express φ' in the modal basis: modal amplitudes q N independent equations for each modal amplitude
Modal solution to the IVP: stability envelope oscillations coefficients from initial condition Linear instability if at least one eigenvalue with sr>0 (unbounded growth of a fundamental solution) Linear stability if ALL eigenvalues have sr<0 Linear stability analysis: given L compute its eigenvalues and check their real parts
THE METHOD OF NORMAL MODES FOR LINEAR PDEs: AN EXAMPLE
The case of a `parallel' unconfined system 1D reaction-diffusion eqn BC: solution bounded as x functions for which Lψ=sψ BUT they remain bounded only if pr=0 eigenfunctions = Fourier modes of (real) wavenumber k uncountable infinity of modes (derives from translational invariance of the system) eigenvalues found replacing the eigenfunction in in Lψ=sψ dispersion relation relating the complex temporal eigenvalue to the wavenumber k and the control parameter r
Modal decomposition for unconfined parallel flows modal decomposition: sum integral on index k modal decomposition = inverse Fourier transform modal decomposition usual procedure usual solution: Linear instability: if at least one k exists for which sr(k)>0 Linear stability: if sr(k)<0 for all k
Typical stability plots: growth rate growth rate vs. wavenumber for selected values of r maximum growth rate sr,max from dispersion relation: sr = r - k² most amplified k:,kmax unstable waveband for r=1 first instability appears at rc,kc growth rate wavenumber
Typical stability plots: neutral curve neutral curve sr=0: separates regions with positive growth rate from regions of negative growth rate in the r-k plane r=1 unstable waveband stable region (sr<0) neutral curve rneut(k) unstable region (sr>0) where sr=0. From the dispersion relation: sr = r - k² enforcing sr=0 rneut(k) = k² critical point rc,kc defined as min k[rneut(k)]
MODAL STABILITY OF PARALLEL SHEAR FLOWS
Parallel incompressible shear flows Some flows of interest are exactly parallel: Other are weakly non parallel 'local' analysis at some x Will consider parallel basic flows U = {U(y),0,0} Control parameter: Reynolds number = ratio btw viscous time scale δ²/ν and time scale of shear δ/ U Re = ΔU δ / ν max velocity variation length scale of shear region kinematic viscosity
Linearized Navier-Stokes eqns. linearized Navier-Stokes equations usually transformed into v-η form (exploiting div u'=0): wall-normal velocity v wall-normal vorticity η
Orr-Sommerfeld-Squire system consider Fourier modes in x-z (unconfined homogeneous direction)s; D := d/dy replace and find the Orr-Sommerfeld-Squire system Orr-Sommerfeld operator Squire operator eigenvalues eigenfunctions found numerically solving the problem in the y direction
Fundamental results for parallel shear flows U(y) Squire (viscous) theorem: The critical mode, becoming unstable at the lowest Reynolds number, is two-dimensional Squire (inviscid) theorem: In the inviscid case, given an unstable 3D mode, a 2D more unstable mode can always be found Confirmed by experimental observations consider 2D perturbations in modal stability analyses Rayleigh inflection point theorem: A necessary condition for U(y) of class at least C² to be unstable to 2D inviscid perturbations satisfying v(yb ) = 0, v(ya ) = 0 is that U(y) admits at least one inflection point in ] yb, ya [ Inviscid instabilities only for inflectional profiles!
typical free shear flows (no walls) inviscid instabilities allowed by inflectional profiles instability develops on short (shear) time scales typical wall-bounded flows no inflection points may have viscous instabilities (Tollmien Schlichting) vanishing as Re develop on long (viscous time scales) e e figures: from Manneville, 2010 Typical neutral curves
TRANSITION TO TURBULENCE IN SHEAR FLOWS
Transition to turbulence in mixing layers Side view (xy) Transition in a spatial mixing layer (Brown & Roshko J. Flud Mech. 1974) Top view (xz) 2D fronts in transition 3D structures appear later & important in turbulent flow Similar trends observed for modal instabilities of jets, wakes,
`Classical' transition scenario in boundary layers scenario observed in the presence of very low external noise levels side view (x-y) developed turbulence Re increases downstream Primary instability of U(y) profile 2D Tollmien-Schlichting waves exponential amplification of unstable 2D TS waves top view (x-z) TS waves reach critical amplitude ~ 1% Ue secondary instability to 3D waves amplification of 3D waves new instability figure: from DNS of Schlatter, KTH
HOWEVER... some flows do not follow this scenario: Plane Couette & circular pipe flow: linearly stable for all Re but become turbulent at finite Re Plane Poiseuille flow: transition almost always observed for Re well below Rec
Bypass transition in boundary layers Transition for Re < Rec No evidence of 2D TS waves subcritical Purely nonlinear mechanism? Matsubara & Alfredsson,2001 Scenario observed in noisy environments Structures observed before transition: streaks Streaks: uniform in z / ~ periodic in x TS waves: ~periodic in x / uniform in z
in the 1990-2000': a lot about bypass transition understood reconsidering classical linear stability analysis
ENERGY AMPLIFICATION IN LINEARLY STABLE SYSTEMS
Reminder: 'classical' linear stability analysis perturbations state vector Linear IVP depends on Re & basic flow Linear asymptotic stability requirement Standard modal stability analysis: * Compute the spectrum of L * Linear asymptotic stability if spectrum is in the stable complex half-plane
Results apply when t What happens at finite times? Can φ' ² become large during transients? Not forbidden by asymptotic stability... but does energy actually grow? How much? Find worst case disturbances optimal growth
Optimal energy growth in the IVP Formal solution of linear IVP Optimal energy growth: propagator from t=0 to actual t initial condition input- output definition of operator norm * to each t corresponds a different G & optimal IC * G(t): envelope of all energy gain curves
No energy growth if L is stable & normal Simple 2D case with real eigenvalues-eigenvectors: φ'(t) = q1(0) Ψ(1) exp(s(1)t)+q2(0) Ψ(2) exp(s(2)t) Assume stable eigenvalues with s(2) the most stable and orthogonal eigenvectors Ψ(j). Choose q1(0)=-1, q2(0)=1 φ'(0) = Ψ(2) - Ψ(1) φ'0 = Ψ(2) - Ψ(1) Ψ(2) exp(s(2)t) φ'(t) Ψ(2) Ψ(1) Ψ(1) exp(s(1)t) the norm of φ' can only decrease monotonously
Non-normality & transient growth If stable eigenvalues & orthogonal eigenvectors energy growth impossible G=1 What happens if the eigenvectors are non-orthogonal? (i.e. if L is non-normal: LL+ L+L) Ψ(2) Ψ(2) exp(s(2)t) Ψ(1) Ψ(1) exp(s(1)t) Necessary condition for transient growth (G>1): non-normal L (non-orthogonal eigenvectors) φ'(t) φ'(t) transiently grows & changes direction
Do-it-yourself: a simple 2x2 example non normal matrix ~ Orr-Sommerfeld-Squire system eigenvalues: -1/Re and -3/Re linearly stable eigenvectors: non-orthogonal with angle decreasing with Re G(t) = etl computed in a few matlab (or octave!) lines: Reynolds=20 L=[-1/Reynolds, 0 ; 1, -3/Reynolds] t=linspace(0,60,60); for j=1:60 P=expm(t(j)*L); G(j)=(norm(P))^2; end plot(t,g);
2x2 example: scaling of growth with Re Gmax ~ Re² tmax ~ Re same Re scalings as genuine Navier-Stokes case optimal perturbations associated to Gmax: optimal IC first component ~ v response at tmax: second component ~ η
Optimal amplification in forced responses linear forced system complex harmonic forcing/response (L assumed stable) resolvent operator also called pseudospectrum infinite if ζ=eigenvalue optimal amplification of forcing energy
Response to harmonic forcing harmonic forcing: ζ = i ω resolvent norm computed for 2x2 toy model: Reynolds=20 L=[-1/Reynolds, 0 ; 1, -3/Reynolds] freq=linspace(-2,2,40); for j=1:40 Resol=inv(freq(j)*eye(2)-L); R(j)=(norm(Resol))^2; end plot(t,r); non-normality large excitability far from resonance! reference curve: response that would be obtained with a normal matrix with same eigenvalues
SHEAR FLOWS: STREAKS, VORTICES & SELF-SUSTAINED PROCESSES
streamwise streaks ~ η streamwise vortices ~ v high speed streak low speed streak As high speed low speed spanwise streamwise wall normal Vortices & streaks: the lift-up effect spanwise
Tentative explanation: the self-sustained process lift-up effect: selection of amplified spanwise scales λz (waveband β) selection of unstable streamwise scales λx (waveband α) SSP selfsustained process Figure from Hamilton et al..jfm 1995
Streaks in wall-bounded turbulent flows SSP mechanism active at small scale near walls SSP active also at large scale (not induced by small scale) Kline et al. JFM 1967 Hwang & Cossu,Phys. Rev. Lett. 2010.
USING EXCITABILITY TO MANIPULATE THE FLOW
streaks & flow control stable streaks (below critical amplitude): strongly modify the basic flow 2D U(y) 3D U(y,z) can be forced with low O(1/Re²) input energy transient growth efficiently used to modify basic flow used to stabilize the flow * lift-up effect: O(Re²) actuator energy amplifier * kill one instability with another = vaccination
Forcing streaks with roughness elements
Experimental test of flow vaccination
Transition delay with forced streaks No streaks forced Streaks forced Basic flow 0 mv 0 mv 205 mv 450 mv Basic flow + tripping (unsteady) forcing Fransson, Talamelli, Brandt & Cossu, Phys. Rev. Lett. 2006
general idea: use optimal energy amplification to manipulate flows idea extended to coherent non-normal amplification in turbulent flows can probably be extended to MHD applications
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